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Theorem noinfbnd1lem4 27690

Description: Lemma for noinfbnd1 27693. If 𝑈 is a prolongment of 𝑇 and in 𝐵, then (𝑈‘dom 𝑇) is not undefined. (Contributed by Scott Fenton, 9-Aug-2024.)

**Hypothesis**
Ref	Expression
noinfbnd1.1	⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1_o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))

**Assertion**
Ref	Expression
noinfbnd1lem4	⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → (𝑈‘dom 𝑇) ≠ ∅)

Distinct variable groups: 𝐵,𝑔,𝑢,𝑣,𝑥,𝑦 𝑣,𝑈 𝑔,𝑉 𝑥,𝑈,𝑦 Allowed substitution hints: 𝑇(𝑥,𝑦,𝑣,𝑢,𝑔) 𝑈(𝑢,𝑔) 𝑉(𝑥,𝑦,𝑣,𝑢)

Proof of Theorem noinfbnd1lem4 Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1192	. . . . . . . 8 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → ¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥)
2		simpl2 1193	. . . . . . . 8 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉))
3		simprl 770	. . . . . . . 8 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → 𝑤 ∈ 𝐵)
4		simpl3 1194	. . . . . . . . 9 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇))
5		simp2l 1200	. . . . . . . . . . . . 13 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → 𝐵 ⊆ No )
6	5	sselda 3958	. . . . . . . . . . . 12 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ No )
7		simp3l 1202	. . . . . . . . . . . . . 14 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → 𝑈 ∈ 𝐵)
8	5, 7	sseldd 3959	. . . . . . . . . . . . 13 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → 𝑈 ∈ No )
9	8	adantr 480	. . . . . . . . . . . 12 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ 𝑤 ∈ 𝐵) → 𝑈 ∈ No )
10		sltso 27640	. . . . . . . . . . . . 13 ⊢ <s Or No
11		soasym 5594	. . . . . . . . . . . . 13 ⊢ (( <s Or No ∧ (𝑤 ∈ No ∧ 𝑈 ∈ No )) → (𝑤 <s 𝑈 → ¬ 𝑈 <s 𝑤))
12	10, 11	mpan 690	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ No ∧ 𝑈 ∈ No ) → (𝑤 <s 𝑈 → ¬ 𝑈 <s 𝑤))
13	6, 9, 12	syl2anc 584	. . . . . . . . . . 11 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ 𝑤 ∈ 𝐵) → (𝑤 <s 𝑈 → ¬ 𝑈 <s 𝑤))
14	13	impr 454	. . . . . . . . . 10 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → ¬ 𝑈 <s 𝑤)
15	3, 14	jca 511	. . . . . . . . 9 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → (𝑤 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑤))
16		noinfbnd1.1	. . . . . . . . . 10 ⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1_o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
17	16	noinfbnd1lem2 27688	. . . . . . . . 9 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ ((𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑤 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑤))) → (𝑤 ↾ dom 𝑇) = 𝑇)
18	1, 2, 4, 15, 17	syl112anc 1376	. . . . . . . 8 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → (𝑤 ↾ dom 𝑇) = 𝑇)
19	16	noinfbnd1lem3 27689	. . . . . . . 8 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑤 ∈ 𝐵 ∧ (𝑤 ↾ dom 𝑇) = 𝑇)) → (𝑤‘dom 𝑇) ≠ 1_o)
20	1, 2, 3, 18, 19	syl112anc 1376	. . . . . . 7 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → (𝑤‘dom 𝑇) ≠ 1_o)
21	20	neneqd 2937	. . . . . 6 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → ¬ (𝑤‘dom 𝑇) = 1_o)
22	21	expr 456	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ 𝑤 ∈ 𝐵) → (𝑤 <s 𝑈 → ¬ (𝑤‘dom 𝑇) = 1_o))
23		imnan 399	. . . . 5 ⊢ ((𝑤 <s 𝑈 → ¬ (𝑤‘dom 𝑇) = 1_o) ↔ ¬ (𝑤 <s 𝑈 ∧ (𝑤‘dom 𝑇) = 1_o))
24	22, 23	sylib 218	. . . 4 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ 𝑤 ∈ 𝐵) → ¬ (𝑤 <s 𝑈 ∧ (𝑤‘dom 𝑇) = 1_o))
25	24	nrexdv 3135	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → ¬ ∃𝑤 ∈ 𝐵 (𝑤 <s 𝑈 ∧ (𝑤‘dom 𝑇) = 1_o))
26		breq2 5123	. . . . . . 7 ⊢ (𝑥 = 𝑈 → (𝑦 <s 𝑥 ↔ 𝑦 <s 𝑈))
27	26	rexbidv 3164	. . . . . 6 ⊢ (𝑥 = 𝑈 → (∃𝑦 ∈ 𝐵 𝑦 <s 𝑥 ↔ ∃𝑦 ∈ 𝐵 𝑦 <s 𝑈))
28		simpl1 1192	. . . . . . 7 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → ¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥)
29		dfral2 3088	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑦 <s 𝑥 ↔ ¬ ∃𝑥 ∈ 𝐵 ¬ ∃𝑦 ∈ 𝐵 𝑦 <s 𝑥)
30		ralnex 3062	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ↔ ¬ ∃𝑦 ∈ 𝐵 𝑦 <s 𝑥)
31	30	rexbii 3083	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ↔ ∃𝑥 ∈ 𝐵 ¬ ∃𝑦 ∈ 𝐵 𝑦 <s 𝑥)
32	29, 31	xchbinxr 335	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑦 <s 𝑥 ↔ ¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥)
33	28, 32	sylibr 234	. . . . . 6 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑦 <s 𝑥)
34		simpl3l 1229	. . . . . 6 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → 𝑈 ∈ 𝐵)
35	27, 33, 34	rspcdva 3602	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → ∃𝑦 ∈ 𝐵 𝑦 <s 𝑈)
36		breq1 5122	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 <s 𝑈 ↔ 𝑤 <s 𝑈))
37	36	cbvrexvw 3221	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 𝑦 <s 𝑈 ↔ ∃𝑤 ∈ 𝐵 𝑤 <s 𝑈)
38	35, 37	sylib 218	. . . 4 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → ∃𝑤 ∈ 𝐵 𝑤 <s 𝑈)
39		simpl2l 1227	. . . . . . . . . 10 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → 𝐵 ⊆ No )
40	39	adantr 480	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → 𝐵 ⊆ No )
41		simprl 770	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → 𝑤 ∈ 𝐵)
42	40, 41	sseldd 3959	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → 𝑤 ∈ No )
43	34	adantr 480	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → 𝑈 ∈ 𝐵)
44	40, 43	sseldd 3959	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → 𝑈 ∈ No )
45		simpl2 1193	. . . . . . . . . . 11 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉))
46	16	noinfno 27682	. . . . . . . . . . 11 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) → 𝑇 ∈ No )
47	45, 46	syl 17	. . . . . . . . . 10 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → 𝑇 ∈ No )
48	47	adantr 480	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → 𝑇 ∈ No )
49		nodmon 27614	. . . . . . . . 9 ⊢ (𝑇 ∈ No → dom 𝑇 ∈ On)
50	48, 49	syl 17	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → dom 𝑇 ∈ On)
51		simpll1 1213	. . . . . . . . . 10 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → ¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥)
52		simpll2 1214	. . . . . . . . . 10 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉))
53		simpll3 1215	. . . . . . . . . 10 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇))
54		simprr 772	. . . . . . . . . . . 12 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → 𝑤 <s 𝑈)
55	42, 44, 12	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → (𝑤 <s 𝑈 → ¬ 𝑈 <s 𝑤))
56	54, 55	mpd 15	. . . . . . . . . . 11 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → ¬ 𝑈 <s 𝑤)
57	41, 56	jca 511	. . . . . . . . . 10 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → (𝑤 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑤))
58	51, 52, 53, 57, 17	syl112anc 1376	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → (𝑤 ↾ dom 𝑇) = 𝑇)
59		simpl3r 1230	. . . . . . . . . 10 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → (𝑈 ↾ dom 𝑇) = 𝑇)
60	59	adantr 480	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → (𝑈 ↾ dom 𝑇) = 𝑇)
61	58, 60	eqtr4d 2773	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → (𝑤 ↾ dom 𝑇) = (𝑈 ↾ dom 𝑇))
62		simplr 768	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → (𝑈‘dom 𝑇) = ∅)
63		nogt01o 27660	. . . . . . . 8 ⊢ (((𝑤 ∈ No ∧ 𝑈 ∈ No ∧ dom 𝑇 ∈ On) ∧ ((𝑤 ↾ dom 𝑇) = (𝑈 ↾ dom 𝑇) ∧ 𝑤 <s 𝑈) ∧ (𝑈‘dom 𝑇) = ∅) → (𝑤‘dom 𝑇) = 1_o)
64	42, 44, 50, 61, 54, 62, 63	syl321anc 1394	. . . . . . 7 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈)) → (𝑤‘dom 𝑇) = 1_o)
65	64	expr 456	. . . . . 6 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ 𝑤 ∈ 𝐵) → (𝑤 <s 𝑈 → (𝑤‘dom 𝑇) = 1_o))
66	65	ancld 550	. . . . 5 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ 𝑤 ∈ 𝐵) → (𝑤 <s 𝑈 → (𝑤 <s 𝑈 ∧ (𝑤‘dom 𝑇) = 1_o)))
67	66	reximdva 3153	. . . 4 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → (∃𝑤 ∈ 𝐵 𝑤 <s 𝑈 → ∃𝑤 ∈ 𝐵 (𝑤 <s 𝑈 ∧ (𝑤‘dom 𝑇) = 1_o)))
68	38, 67	mpd 15	. . 3 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → ∃𝑤 ∈ 𝐵 (𝑤 <s 𝑈 ∧ (𝑤‘dom 𝑇) = 1_o))
69	25, 68	mtand 815	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → ¬ (𝑈‘dom 𝑇) = ∅)
70	69	neqned 2939	1 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → (𝑈‘dom 𝑇) ≠ ∅)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 {cab 2713 ≠ wne 2932 ∀wral 3051 ∃wrex 3060 ∪ cun 3924 ⊆ wss 3926 ∅c0 4308 ifcif 4500 {csn 4601 ⟨cop 4607 class class class wbr 5119 ↦ cmpt 5201 Or wor 5560 dom cdm 5654 ↾ cres 5656 Oncon0 6352 suc csuc 6354 ℩cio 6482 ‘cfv 6531 ℩crio 7361 1_oc1o 8473 No csur 27603 <s cslt 27604

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-ord 6355 df-on 6356 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-fo 6537 df-fv 6539 df-riota 7362 df-1o 8480 df-2o 8481 df-no 27606 df-slt 27607 df-bday 27608

This theorem is referenced by: noinfbnd1lem5 27691 noinfbnd1lem6 27692

W3C validator